TÀI LIỆU DẠY TOÁN BẰNG TIẾNG ANH CỰC HAY

Grade-Three Chapter of the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve Adopted by the California State Board of Education, November 2013 Published by the California Department of Education Sacramento, 2015 Grade Three K I n grade three, students continue to build upon their mathematical foundation as they focus on the operations of multiplication and division and the concept of fractions as numbers In previous grades, students developed an understanding of place value and used methods based on place value to add and subtract within 1000 They developed fluency with addition and subtraction within 100 and laid a foundation for understanding multiplication based on equal groups and the array model Students also worked with standard units to measure length and described attributes of geometric shapes (adapted from Charles A Dana Center 2012) Critical Areas of Instruction In grade three, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division, as well as strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with a numerator of 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes (National Governors Association Center for Best Practices, Council of Chief State School Officers [NGA/CCSSO] 2010j) Students also work toward fluency with addition and subtraction within 1000 and multiplication and division within 100 By the end of grade three, students know all products of two one-digit numbers from memory California Mathematics Framework Grade Three 157 Standards for Mathematical Content The Standards for Mathematical Content emphasize key content, skills, and practices at each grade level and support three major principles: • Focus—Instruction is focused on grade-level standards • Coherence—Instruction should be attentive to learning across grades and to linking major topics within grades • Rigor—Instruction should develop conceptual understanding, procedural skill and fluency, and application Grade-level examples of focus, coherence, and rigor are indicated throughout the chapter The standards not give equal emphasis to all content for a particular grade level Cluster headings can be viewed as the most effective way to communicate the focus and coherence of the standards Some clusters of standards require a greater instructional emphasis than others based on the depth of the ideas, the time needed to master those clusters, and their importance to future mathematics or the later demands of preparing for college and careers Table 3-1 highlights the content emphases at the cluster level for the grade-three standards The bulk of instructional time should be given to “Major” clusters and the standards within them, which are indicated throughout the text by a triangle symbol ( ) However, standards in the “Additional/Supporting” clusters should not be neglected; to so would result in gaps in students’ learning, including skills and understandings they may need in later grades Instruction should reinforce topics in major clusters by using topics in the additional/ supporting clusters and including problems and activities that support natural connections between clusters Teachers and administrators alike should note that the standards are not topics to be checked off after being covered in isolated units of instruction; rather, they provide content to be developed throughout the school year through rich instructional experiences presented in a coherent manner (adapted from Partnership for Assessment of Readiness for College and Careers [PARCC] 2012) Table 3-1 Grade Three Cluster-Level Emphases Operations and Algebraic Thinking 3.OA Major Clusters • • • • Represent and solve problems involving multiplication and division (3.OA.1–4 ) Understand properties of multiplication and the relationship between multiplication and division (3.OA.5–6 ) Multiply and divide within 100 (3.OA.7 ) Solve problems involving the four operations, and identify and explain patterns in arithmetic (3.OA.8–9 ) Number and Operations in Base Ten 3.NBT Additional/Supporting Clusters • Use place-value understanding and properties of operations to perform multi-digit arithmetic (3.NBT.1–3) Number and Operations—Fractions 3.NF Major Clusters • Develop understanding of fractions as numbers (3.NF.1–3 ) Measurement and Data 3.MD Major Clusters • • Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects (3.MD.1–2 ) Geometric measurement: understand concepts of area and relate area to multiplication and to addition (3.MD.5–7 ) Additional/Supporting Clusters • • Represent and interpret data (3.MD.3–4) Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures (3.MD.8) Geometry 3.G Additional/Supporting Clusters • Reason with shapes and their attributes (3.G.1–2) Explanations of Major and Additional/Supporting Cluster-Level Emphases Major Clusters ( ) — Areas of intensive focus where students need fluent understanding and application of the core concepts These clusters require greater emphasis than others based on the depth of the ideas, the time needed to master them, and their importance to future mathematics or the demands of college and career readiness Additional Clusters — Expose students to other subjects; may not connect tightly or explicitly to the major work of the grade Supporting Clusters — Designed to support and strengthen areas of major emphasis Note of caution: Neglecting material, whether it is found in the major or additional/supporting clusters, will leave gaps in students’ skills and understanding and will leave students unprepared for the challenges they face in later grades Adapted from Smarter Balanced Assessment Consortium 2011, 83 Connecting Mathematical Practices and Content The Standards for Mathematical Practice (MP) are developed throughout each grade and, together with the content standards, prescribe that students experience mathematics as a rigorous, coherent, useful, and logical subject The MP standards represent a picture of what it looks like for students to understand and mathematics in the classroom and should be integrated into every mathematics lesson for all students Although the description of the MP standards remains the same at all grade levels, the way these standards look as students engage with and master new and more advanced mathematical ideas does change Table 3-2 presents examples of how the MP standards may be integrated into tasks appropriate for students in grade three (Refer to the Overview of the Standards Chapters for a description of the MP standards.) Table 3-2 Standards for Mathematical Practice—Explanation and Examples for Grade Three Standards for Mathematical Practice MP.1 Make sense of problems and persevere in solving them MP.2 Reason abstractly and quantitatively Explanation and Examples In third grade, mathematically proficient students know that doing mathematics involves solving problems and discussing how they solved them Students explain to themselves the meaning of a problem and look for ways to solve it Students may use concrete objects, pictures, or drawings to help them conceptualize and solve problems such as these: “Jim purchased packages of muffins Each package contained muffins How many muffins did Jim purchase?”; or “Describe another situation where there would be groups of or × ” Students may check their thinking by asking themselves, “Does this make sense?” Students listen to other students’ strategies and are able to make connections between various methods for a given problem Students recognize that a number represents a specific quantity They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities For example, students apply their understanding of the meaning of the equal sign as “the same as” to interpret an equation with an unknown When given × — = 40 , they might think: • groups of some number is the same as 40 • times some number is the same as 40 • I know that groups of 10 is 40, so the unknown number is 10 • The missing factor is 10, because times 10 equals 40 To reinforce students’ reasoning and understanding, teachers might ask, “How you know?” or “What is the relationship between the quantities?” MP.3 Construct viable arguments and critique the reasoning of others Students may construct arguments using concrete referents, such as objects, pictures, and drawings They refine their mathematical communication skills as they participate in mathematical discussions that the teacher facilitates by asking questions such as “How did you get that?” and “Why is that true?” Students explain their thinking to others and respond to others’ thinking For example, after investigating patterns on the hundreds chart, students might explain why the pattern makes sense Table 3-2 (continued) Standards for Mathematical Practice MP.4 Model with mathematics MP.5 Use appropriate tools strategically MP.6 Attend to precision MP.7 Look for and make use of structure MP.8 Look for and express regularity in repeated reasoning Explanation and Examples Students represent problem situations in multiple ways using numbers, words (mathematical language), objects, and math drawings They might also represent a problem by acting it out or by creating charts, lists, graphs, or equations For example, students use various contexts and a variety of models (e.g., circles, squares, rectangles, fraction bars, and number lines) to represent and develop understanding of fractions Students use models to represent both equations and story problems and can explain their thinking They evaluate their results in the context of the situation and reflect on whether the results make sense Students should be encouraged to answer questions such as “What math drawing or diagram could you make and label to represent the problem?” or “What are some ways to represent the quantities?” Mathematically proficient students consider the available tools (including drawings or estimation) when solving a mathematical problem and decide when particular tools might be helpful For instance, they may use graph paper to find all the possible rectangles that have a given perimeter They compile the possibilities into an organized list or a table and determine whether they have all the possible rectangles Students should be encouraged to answer questions (e.g., “Why was it helpful to use ?”) Students develop mathematical communication skills as they use clear and precise language in their discussions with others and in their own reasoning They are careful to specify units of measure and to state the meaning of the symbols they choose For instance, when calculating the area of a rectangle they record the answer in square units Students look closely to discover a pattern or structure For instance, students use properties of operations (e.g., commutative and distributive properties) as strategies to multiply and divide Teachers might ask, “What you notice when ?” or “How you know if something is a pattern?” Students notice repetitive actions in computations and look for “shortcut” methods For instance, students may use the distributive property as a strategy to work with products of numbers they know to solve products they not know For example, to find the product of × , students might decompose into and and then multiply × and × to arrive at 40 + 16 , or 56 Third-grade students continually evaluate their work by asking themselves, “Does this make sense?” Students should be encouraged to answer questions such as “What is happening in this situation?” or “What predictions or generalizations can this pattern support?” Adapted from Arizona Department of Education (ADE) 2010 and North Carolina Department of Public Instruction (NCDPI) 2013b Standards-Based Learning at Grade Three The following narrative is organized by the domains in the Standards for Mathematical Content and highlights some necessary foundational skills from previous grade levels It also provides exemplars to explain the content standards, highlight connections to Standards for Mathematical Practice (MP), and demonstrate the importance of developing conceptual understanding, procedural skill and fluency, and application A triangle symbol ( ) indicates standards in the major clusters (see table 3-1) California Mathematics Framework Grade Three 161 Domain: Operations and Algebraic Thinking In kindergarten through grade two, students focused on developing an understanding of addition and subtraction Beginning in grade three, students focus on concepts, skills, and problem solving for multiplication and division Students develop multiplication strategies, make a shift from additive to multiplicative reasoning, and relate division to multiplication Third-grade students become fluent with multiplication and division within 100 This work will continue in grades four and five, preparing the way for work with ratios and proportions in grades six and seven (adapted from the University of Arizona Progressions Documents for the Common Core Math Standards [UA Progressions Documents] 2011a and PARCC 2012) Multiplication and division are new concepts in grade three, and meeting fluency is a major portion of students’ work (see 3.OA.7 ) Reaching fluency will take much of the year for many students These skills and the understandings that support them are crucial; students will rely on them for years to come as they learn to multiply and divide with multi-digit whole numbers and to add, subtract, multiply, and divide with rational numbers There are many patterns to be discovered by exploring the multiples of numbers Examining and articulating these patterns is an important part of the mathematical work on multiplication and division Practice—and, if necessary, extra support—should continue all year for those students who need it to attain fluency This practice can begin with the easier multiplication and division problems while the pattern work is occurring with more difficult numbers (adapted from PARCC 2012) Relating and practicing multiplication and division problems involving the same number (e.g., the 4s) may be helpful Operations and Algebraic Thinking 3.OA Represent and solve problems involving multiplication and division Interpret products of whole numbers, e.g., interpret × as the total number of objects in groups of objects each For example, describe a context in which a total number of objects can be expressed as × Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ as the number of objects in each share when 56 objects are partitioned equally into shares, or as a number of shares when 56 objects are partitioned into equal shares of objects each For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem Determine the unknown whole number in a multiplication or division equation relating three whole numbers For example, determine the unknown number that makes the equation true in each of the equations × ? = 48 , = ÷ , × = ? A critical area of instruction is to develop student understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models (NGA/CCSSO 2010c) Multiplication and division are new concepts in grade three Initially, See glossary, table GL-5 162 Grade Three California Mathematics Framework students need opportunities to develop, discuss, and use efficient, accurate, and generalizable methods to compute The goal is for students to use general written methods for multiplication and division that students can explain and understand (e.g., using visual models or place-value language) The general written methods should be variations of the standard algorithms Reaching fluency with these operations requires students to use variations of the standard algorithms without visual models, and this could take much of the year for many students Students recognize multiplication as finding the total number of objects in a particular number of equal-sized groups (3.OA.1 ) Also, students recognize division in two different situations: partitive division (also referred to as fair-share division), which requires equal sharing (e.g., how many are in each group?); and quotitive division (or measurement division), which requires determining how many groups (e.g., how many groups can you make?) [3.OA.2 ] These two interpretations of division have important uses later, when students study division of fractions, and both interpretations should be explored as representations of division In grade three, teachers should use the terms number of shares or number of groups with students rather than partitive or quotitive Multiplication of Whole Numbers 3.OA.1 Note that the standards define multiplication of whole numbers a × b as finding the total number of objects in a groups of b objects Example: There are bags of apples on the table There are apples in each bag How many apples are there altogether? Partitive Division (also known as Fair-Share or Group Size Unknown Division) 3.OA.2 The number of groups or shares to be made is known, but the number of objects in (or size of) each group or share is unknown Example: There are 12 apples on the counter If you are sharing the apples equally among bags, how many apples will go in each bag? Quotitive Division (also known as Measurement or Number of Groups Unknown Division) 3.OA.2 The number of objects in (or size of) each group or share is known, but the number of groups or shares is unknown Example: There are 12 apples on the counter If you put apples in each bag, how many bags will you fill? Students are exposed to related terminology for multiplication (factor and product) and division (quotient, dividend, divisor, and factor) They use multiplication and division within 100 to solve word problems (3.OA.3 ) in situations involving equal groups, arrays, and measurement quantities Note that although “repeated addition” can be used in some cases as a strategy for finding whole-number products, repeated addition should not be misconstrued as the meaning of multiplication The intention of the standards in grade three is to move students beyond additive thinking to multiplicative thinking The three most common types of multiplication and division word problems for this grade level are summarized in table 3-3 California Mathematics Framework Grade Three 163 Table 3-3 Types of Multiplication and Division Problems (Grade Three) Equal Groups Arrays, Area Compare Unknown Product Group Size Unknown2 Number of Groups Unknown3 =? ? = 18 and 18 ÷ = ? ? × = 18 and 18 ÷ = ? There are bags with plums in each bag How many plums are there altogether? If 18 plums are shared equally and packed into bags, then how many plums will be in each bag? Measurement example You need lengths of string, each inches long How much string will you need altogether? Measurement example You have 18 inches of string, which you will cut into equal pieces How long will each piece of string be? Measurement example You have 18 inches of string, which you will cut into pieces that are inches long How many pieces of string will you have? There are rows of apples with apples in each row How many apples are there? If 18 apples are arranged into equal rows, how many apples will be in each row? If 18 apples are arranged into equal rows of apples, how many rows will there be? Area example What is the area of a rectangle that measures centimeters by centimeters? Area example A rectangle has an area of 18 square centimeters If one side is centimeters long, how long is a side next to it? Area example A rectangle has an area of 18 square centimeters If one side is centimeters long, how long is a side next to it? If 18 plums are to be packed, with plums to a bag, then how many bags are needed? Grade-three students not solve multiplicative “compare” problems; these problems are introduced in grade four (with whole-number values) and also appear in grade five (with unit fractions) General a b =? a ? = p and p ÷ a = ? ? b = p and p ÷ b = ? Source: NGA/CCSSO 2010d A nearly identical version of this table appears in the glossary (table GL-5) In grade three, students focus on equal groups and array problems Multiplicative-compare problems are introduced in grade four The more difficult problem types include “Group Size Unknown” (3 ? = 18 or 18 ÷ = ) or “Number of Groups Unknown” ( ? = 18, 18 ÷ = ) To solve problems, students determine the unknown whole number in a multiplication or division equation relating three whole numbers (3.OA.4 ) Students use numbers, words, pictures, physical objects, or equations to represent problems, explain their thinking, and show their work (MP.1, MP.2, MP.4, MP.5) These problems ask the question, “How many in each group?” The problem type is an example of partitive or fair-share division These problems ask the question, “How many groups?” The problem type is an example of quotitive or measurement division 164 Grade Three California Mathematics Framework Example: Number of Groups Unknown 3.OA.4 Molly the zookeeper has 24 bananas to feed the monkeys Each monkey needs to eat bananas How many monkeys can Molly feed? Solution: ? = 24 Students might draw on the remembered product × = 24 to say that the related quotient is Alternatively, they might draw on other known products—for example, if × = 20 is known, then since 20 + = 24 , one more group of will give the desired factor ( + = ) Or, knowing that = 12 and 12 + 12 = 24 , students might reason that the desired factor is + = Any of these methods (or others) might be supported by a representational drawing that shows the equal groups in the situation Operations and Algebraic Thinking 3.OA Understand properties of multiplication and the relationship between multiplication and division Apply properties of operations as strategies to multiply and divide.4 Examples: If × = 24 is known, then × = 24 is also known (Commutative property of multiplication.) × × can be found by = 15 , then 15 × = 30 , or by × = 10 , then 10 = 30 (Associative property of multiplication.) Knowing that × = 40 and × = 16 , one can find × as × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56 (Distributive property.) Understand division as an unknown-factor problem For example, find 32 ÷ by finding the number that makes 32 when multiplied by In grade three, students apply properties of operations as strategies to multiply and divide (3.OA.5 ) Third-grade students not need to use the formal terms for these properties Students use increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors By comparing a variety of solution strategies, students learn about the relationship between multiplication and division Focus, Coherence, and Rigor Arrays can be seen as equal-sized groups where objects are arranged by rows and columns, and they form a major transition to understanding multiplication as finding area (connection to 3.MD.7 ) For example, students can analyze the structure of multiplication and division (MP.7) through their work with arrays (MP.2) and work toward precisely expressing their understanding of the connections between area and multiplication (MP.6) The distributive property is the basis for the standard multiplication algorithm that students can use to fluently multiply multi-digit whole numbers in grade five Third-grade students are introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they not know (MP.2, MP.7) Students need not use formal terms for these properties California Mathematics Framework Grade Three 165 Table 3-4 Connecting to the Standards for Mathematical Practice—Grade Three Standards Addressed Explanation and Examples Task: The Human Fraction Number Line Activity In this activity, the teacher posts a long sheet of paper on MP.2 Students reason quantitatively as they dea wall of the classroom to act as a number line, with termine why a placement was correct or incorrect marked at one end and marked at the other Gathered and assign a fractional value to a distance around the wall, groups of students are given cards with MP.4 Students use the number line model for different-sized fractions indicated on them—for example, fractions Although this is not an application of 4 , , , , —and are asked to locate themselves along mathematics to a real-world situation in the true the number line according to the fractions assigned to sense of modeling, it is an appropriate use of them Depending on the size of the class and the length modeling for the grade level MP.8 Students see repeated reasoning in dividing of the number line, fractions with denominators 2, 3, 4, 6, and may be used The teacher can ask students to up the number line into equal parts (of varied sizes) and form the basis for how they would place explain to each other why their placements are correct fifths, tenths, and other fractions or incorrect, emphasizing that the students with cards marked in fourths, say, have divided the number line into Standards for Mathematical Content a four equal parts Furthermore, a student with the card b 3.NF.1 Understand a fraction as the quantity b is standing in the correct place if he or she represents a formed by part when a whole is partitioned into lengths of size from on the number line b b equal parts; understand a fraction ba as the As a follow-up activity, teachers can give students several quantity formed by a parts of size b 3.NF.2 Understand a fraction as a number on the unit number lines that are marked off into equal parts number line; represent fractions on a number line but that are unlabeled Students are required to fill in the diagram labels on the number lines An example is shown here: a Represent a fraction on a number line b diagram by defining the interval from to Connections to Standards for Mathematical Practice as the whole and partitioning it into b equal parts Recognize that each part has size and b that the endpoint of the part based at locates the number on the number line b b Represent a fraction a on a number line b diagram by marking off a lengths from b Recognize that the resulting interval has size a b and that its endpoint locates the number a on b the number line 3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size b Recognize and generate simple equivalent fractions (e.g., = , = ) Explain why the fractions are equivalent, e.g., by using a visual fraction model Classroom Connections There are several big ideas included in this activity One is that when talking about fractions as points on a number line, the whole is represented by the length or amount of distance from to By requiring students to physically line up in the correct places on the number line, the idea of partitioning this distance into equal parts is emphasized In addition, other students can physically mark off the placement of fractions by starting from and walking the required number of lengths from 0; for example, b with students placed at the locations for sixths, another student can start at and walk off a distance of As an extension, teachers can have students mark off equivalent fraction distances, such as , , and , and can discuss why those fractions represent the same amount Domain: Measurement and Data Measurement and Data 3.MD Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects Tell and write time to the nearest minute and measure time intervals in minutes Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).9 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.10 Students have experience telling and writing time from analog and digital clocks to the hour and half hour in grade one and in five-minute intervals in grade two In grade three, students write time to the nearest minute and measure time intervals in minutes Students solve word problems involving addition and subtraction of time intervals in minutes and represent these problems on a number line (3.MD.1 ) Students begin to understand the concept of continuous measurement quantities, and they add, subtract, multiply, or divide to solve one-step word problems involving such quantities Multiple opportunities to weigh classroom objects and fill containers will help students develop a basic understanding of the size and weight of a liter, a gram, and a kilogram (3.MD.2 ) Focus, Coherence, and Rigor Students’ understanding and work with measuring and estimating continuous measurement quantities, such as liquid volume and mass (3.MD.2 ), are an important context for the fraction arithmetic they will experience in later grade levels Measurement and Data 3.MD Represent and interpret data Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs For example, draw a bar graph in which each square in the bar graph might represent pets Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters Excludes compound units such as cm3 and finding the geometric volume of a container 10 Excludes multiplicative comparison problems (problems involving notions of “times as much”; see glossary, table GL-5) California Mathematics Framework Grade Three 177 In grade three, the most important development in data representation for categorical data is that students draw picture graphs in which each picture represents more than one object, and they draw bar graphs in which the scale uses multiples, so the height of a given bar in tick marks must be multiplied by the scale factor to yield the number of objects in the given category These developments connect with the emphasis on multiplication in this grade (adapted from UA Progressions Documents 2011b) Students draw a scaled pictograph and a scaled bar graph to represent a data set and solve word problems (3.MD.3) Examples 3.MD.3 Students might draw or reference a pictograph with symbols that represent multiple units Number of Books Read Nancy !!!!! !!!!!!!! Note: ! represents books Juan Number of Books Read Students might draw or reference bar graphs to solve related problems 40 35 30 25 20 15 10 Nancy Juan Nancy 10 15 20 25 30 35 40 Number of Books Read Juan Adapted from KATM 2012, 3rd Grade Flipbook Focus, Coherence, and Rigor Pictographs and scaled bar graphs offer a visually appealing context and support major work in the cluster “Represent and solve problems involving multiplication and division” as students solve multiplication and division word problems (3.OA.3 ) Students use their knowledge of fractions and number lines to work with measurement data involving fractional measurement values They generate data by measuring lengths using rulers marked with halves and fourths of an inch and create a line plot to display their findings (3.MD.4) [adapted from UA Progressions Documents 2011b] 178 Grade Three California Mathematics Framework For example, students might use a line plot to display data Number of Objects Measured 4 1 1 Adapted from NCDPI 2013b A critical area of instruction at grade three is for students to develop an understanding of the structure of rectangular arrays and of area measurement Measurement and Data 3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition Recognize area as an attribute of plane figures and understand concepts of area measurement a A square with side length unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area b A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units) Relate area to the operations of multiplication and addition a Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths b Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c Use area models to represent the distributive property in mathematical reasoning d Recognize area as additive Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems Students recognize area as an attribute of plane figures, and they develop an understanding of concepts of area measurement (3.MD.5 ) They discover that a square with a side length of unit, called “a unit square,” is said to have one square unit of area and can be used to measure area California Mathematics Framework Grade Three 179 Students measure areas by counting unit squares (square centimeters, square meters, square inches, square feet, and improvised units) [3.MD.6 ] Students develop an understanding of using square units to measure area by using different-sized square units, filling in an area with the same-sized square units, and then counting the number of square units Students relate the concept of area to the operations of multiplication and addition and show that the area of a rectangle can be found by multiplying the side lengths (3.MD.7 ) Students make sense of these quantities as they learn to interpret measurement of rectangular regions as a multiplicative relationship of the number of square units in a row and the number of rows Students should understand and explain why multiplying the side lengths of a rectangle yields the same measurement of area as counting the number of tiles (with the same unit length) that fill the rectangle’s interior For example, students might explain that one length tells the number of unit squares in a row and the other length tells how many rows there are (adapted from UA Progressions Documents 2012a) Students need opportunities to tile a rectangle with square units and then multiply the side lengths to show that they both give the area For example, to find the area, a student could count the squares or multiply × = 12 10 11 12 The transition from counting unit squares to multiplying side lengths to find area can be aided when students see the progression from multiplication as equal groups to multiplication as a total number of objects in an array, and then see the area of a rectangle as an array of unit squares An example is presented below Students see multiplication as counting objects in equal groups—for example, × as groups of apples: They see the objects arranged in arrays, as in a × array of the same apples: They eventually see that finding area by counting unit squares is like counting an array of objects, where the objects are unit squares Students use area models to represent the distributive property in mathematical reasoning For example, the area of a × figure can be determined by finding the area of a × figure and a × figure and adding the two products 6×5 6×2 Students recognize area as additive and find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts Example 3.MD.7d The standards mention rectilinear figures A rectilinear figure is a polygon whose every angle is a right angle Such figures can be decomposed into rectangles to find their areas Area of rectilinear figure = ? 4×2 2×2 By breaking the figure into two pieces, it becomes easier to see that the area of the figure is + = 12 square units Adapted from NCDPI 2013b Students apply these techniques and understandings to solve real-world problems California Mathematics Framework Grade Three 181 Focus, Coherence, and Rigor The use of area models (3.MD.7 ) also supports multiplicative reasoning, a major focus in grade three in the domain “Operations and Algebraic Thinking” (3.OA.1–9 ) Students must begin work with multiplication and division at or near the start of the school year to allow time for understanding and to develop fluency with these skills Because area models for products are an important part of this process (3.MD.7 ), work on concepts of area (3.MD.5–6 ) should begin at or near the start of the year as well (adapted from PARCC 2012) Measurement and Data 3.MD Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters In grade three, students solve real-world and mathematical problems involving perimeters of polygons (3.MD.8) Students can develop an understanding of the concept of perimeter as they walk around the perimeter of a room, use rubber bands to represent the perimeter of a plane figure with wholenumber side lengths on a geoboard, or trace around a shape on an interactive whiteboard They find the perimeter of objects, use addition to find perimeters, and recognize the patterns that exist when finding the sum of the lengths and widths of rectangles They explain their reasoning to others Given a perimeter and a length or width, students use objects or pictures to find the unknown length or width They justify and communicate their solutions using words, diagrams, pictures, and numbers (adapted from ADE 2010) Domain: Geometry Geometry 3.G Reason with shapes and their attributes Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals) Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that not belong to any of these subcategories Partition shapes into parts with equal areas Express the area of each part as a unit fraction of the whole For example, partition a shape into parts with equal area, and describe the area of each part as 14 of the area of the shape A critical area of instruction at grade three is for students to describe and analyze two-dimensional 182 Grade Three California Mathematics Framework shapes Students compare common geometric shapes (e.g., rectangles and quadrilaterals) based on common attributes, such as four sides (3.G.1) In earlier grades, students informally reasoned about particular shapes through sorting and classifying based on geometric attributes Students also built and drew shapes given the number of faces, number of angles, and number of sides In grade three, students describe properties of two-dimensional shapes in more precise ways, referring to properties that are shared rather than the appearance of individual shapes For example, students could start by identifying shapes with right angles, explain and discuss why the remaining shapes not fit this category, and determine common characteristics of the remaining shapes Students relate their work with fractions to geometry as they partition shapes into parts with equal areas and represent each part as a unit fraction of the whole (3.G.2) Example The figure below was partitioned (divided) into four equal parts Each part is 3.G.2 of the total area of the figure To be prepared for grade-four mathematics, students should be able to demonstrate they have acquired certain mathematical concepts and procedural skills by the end of grade three and have met the fluency expectations for the grade For third-graders, the expected fluencies are to add and subtract within 1000 using strategies and algorithms (3.NBT.2), multiply and divide within 100 using various strategies, and know all products of two one-digit numbers from memory (3.OA.7 ) These fluencies and the conceptual understandings that support them are foundational for work in later grades Of particular importance for grade four are concepts, skills, and understandings needed to represent and solve problems involving multiplication and division (3.OA.1–4 ); understand properties of multiplication and the relationship between multiplication and division (3.OA.5–6 ); multiply and divide within 100 (3.OA.7 ); solve problems involving the four operations and identify and explain patterns in arithmetic (3.OA.8–9 ); develop understanding of fractions as numbers (3.NF.1–3 ); solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects (3.MD.1–2 ); and geometric measurement—concepts of area and relating area to multiplication and to addition (3.MD.5–7 ) Multiplication and Division By the end of grade three, students develop both conceptual understanding and procedural skills of multiplication and division Students are expected to fluently multiply and divide within 100 and to know from memory all of the products of two one-digit numbers (3.OA.7 ) Fluency in multiplication and division within 100 includes understanding and being able to apply strategies such as using mental math, understanding division as an unknown-factor problem, applying the properties of operations, and identifying arithmetic patterns Students also need to understand the relationship between multiplication and division and apply that understanding by using inverse operations to verify the reasonableness of their answers Students with a firm grasp of grade-three multiplication and division can apply their knowledge to interpret, solve, and even compose simple word problems, including problems involving equal groups, arrays, and measurement quantities Fluency in multiplication and division ensures that when students know from memory all of the products of two one-digit numbers, they have an understanding of the two operations—and have not merely learned to produce answers through rote memorization Fractions In grade three, students are formally introduced to fractions as numbers, thus broadening their understanding of the number system Students must understand that fractions are composed of unit fractions; this is essential for their ongoing work with the number system Students must be able to place fractions on a number line and use the number line as a tool to compare fractions and recognize equivalent fractions They should be able to use other visual models to compare fractions Students also must be able to express whole numbers as fractions and place them on a number line It is essential for students to understand that the denominator determines the number of equally sized pieces that make up a whole and the numerator determines how many pieces of the whole are being referred to in the fraction Addition and Subtraction By the end of grade three, students are expected to fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction (3.NBT.2) This fluency is both the culmination of work at previous grade levels and preparation for solving multi-step word problems using all four operations beginning in grade four Students should be able to use more than one strategy to add or subtract and should also be able to relate the strategies they use to a written method California Common Core State Standards for Mathematics Grade Overview Operations and Algebraic Thinking • Represent and solve problems involving multiplication and division • Understand properties of multiplication and the relationship between multiplication and division • • Multiply and divide within 100 Solve problems involving the four operations, and identify and explain patterns in arithmetic Mathematical Practices Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Number and Operations in Base Ten Model with mathematics • Use appropriate tools strategically Use place-value understanding and properties of operations to perform multi-digit arithmetic Number and Operations—Fractions • Develop understanding of fractions as numbers Measurement and Data • Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects • • Represent and interpret data • Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Geometric measurement: understand concepts of area and relate area to multiplication and to addition Geometry • 186 Reason with shapes and their attributes Grade Three California Mathematics Framework Grade Operations and Algebraic Thinking 3.OA Represent and solve problems involving multiplication and division Interpret products of whole numbers, e.g., interpret × as the total number of objects in groups of objects each For example, describe a context in which a total number of objects can be expressed as × Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ as the number of objects in each share when 56 objects are partitioned equally into shares, or as a number of shares when 56 objects are partitioned into equal shares of objects each For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.11 Determine the unknown whole number in a multiplication or division equation relating three whole numbers For example, determine the unknown number that makes the equation true in each of the equations ? = 48 , = ÷ , × = ? Understand properties of multiplication and the relationship between multiplication and division Apply properties of operations as strategies to multiply and divide.12 Examples: If × = 24 is known, then × = 24 is also known (Commutative property of multiplication.) × × can be found by × = 15 , then 15 × = 30 , or by × = 10 , then × 10 = 30 (Associative property of multiplication.) Knowing that × = 40 and × = 16 , one can find × as × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56 (Distributive property.) Understand division as an unknown-factor problem For example, find 32 ÷ 18 by finding the number that makes 32 when multiplied by Multiply and divide within 100 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that × = 40 , one knows 40 ÷ = ) or properties of operations By the end of grade 3, know from memory all products of two one-digit numbers Solve problems involving the four operations, and identify and explain patterns in arithmetic Solve two-step word problems using the four operations Represent these problems using equations with a letter standing for the unknown quantity Assess the reasonableness of answers using mental computation and estimation strategies including rounding.13 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations For example, observe that times a number is always even, and explain why times a number can be decomposed into two equal addends 11 See glossary, table GL-5 12 Students need not use formal terms for these properties 13 This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations) California Mathematics Framework Grade Three 187 Number and Operations in Base Ten 3.NBT Use place-value understanding and properties of operations to perform multi-digit arithmetic.14 Use place-value understanding to round whole numbers to the nearest 10 or 100 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., × 80 , × 60 ) using strategies based on place value and properties of operations 3.NF Number and Operations—Fractions15 Develop understanding of fractions as numbers Understand a fraction b as the quantity formed by part when a whole is partitioned into b equal parts; understand a fraction a b as the quantity formed by a parts of size b Understand a fraction as a number on the number line; represent fractions on a number line diagram a Represent a fraction b on a number line diagram by defining the interval from to as the whole and partitioning it into b equal parts Recognize that each part has size b and that the endpoint of the part based at locates the number b on the number line b Represent a fraction a b on a number line diagram by marking off a lengths b from Recognize that the resulting interval has size a b and that its endpoint locates the number a b on the number line Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line b Recognize and generate simple equivalent fractions, e.g., = , = ) Explain why the fractions are equivalent, e.g., by using a visual fraction model c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers Examples: Express in the form = 31 ; recognize that 61 = ; locate 44 and at the same point of a number line diagram d Compare two fractions with the same numerator or the same denominator by reasoning about their size Recognize that comparisons are valid only when the two fractions refer to the same whole Record the results of comparisons with the symbols >, =, or